The “time-honoured” way to obtain a Weierstrass section is by constructing an adapted pair \((e,h)\)  with \(e \in N\)  and \(h\)  a semisimple element of \(\mathfrak {p}:=\text{Lie} \ P\)  such that \(e\)  is an \(h\)  eigenvector and \(P.e\)  is dense in an irreducible component of \(N\) , that is to say \(e\)  is regular. Then \(V\)  is taken to be an \(h\)  stable complement of \(\mathfrak {p}.e\)  in \(\mathfrak {m}\) . Finally a favourable comparison of degrees of generators and eigenvalues of \(h\)  on V gives the required result. This was the point of view taken in the classical work of Kostant [1]} for \(P=G\) , a simple Lie group. In this he appealed to the entire principal s-triple \((e,h,f)\) . By a further coincidence \(e\)  also generated an orbit of maximal dimension in \(\mathfrak {g}=\text{Lie} \ G\) .
